Transforms and its Applications: UNIT 1: Laplace Transforms
Derivatives of Transform
Laplace Transforms
Laplace Transforms: Example Important Solved Problems with formula, steps, derivation and answer based on Derivatives of Transform.
DERIVATIVES OF
TRANSFORM
Result: Derivatives of Transforms.
If L[f(t)]
= F(s), then L [t f(t)] = ‒ d/ds F(s) = ‒ F '(s)
Proof:
F(s)
= L[f(t)]
L[tf(t)] = −F' (s)
Corollary: If L[f(t)] = F(s), then L [tn
f(1)] = (‒1)n F(n) (s).
Proof:
We
know that, L[tf(t)] = ‒ F '(s)
PROBLEMS BASED ON DERIVATIVES OF TRANSFORM
Example 1: Find L[t sin at]
Solution:
Example 2(a): Find L [t cos at]
Solution:
Example 2(b): L [te‒3t cos 2t]
Solution:
Example 3: Find L [t sin 3t]
Solution:
We
know that, L[tn f(t)] =
(‒1)nF(n)(s)
Example 4: Find L [t2 e‒3t]
Solution:
By
shifting theorem
Example 5: Find L[te‒1 sin t]
Solution:
L[e‒t(t sin t)] = [L [t sin t]]s→s+1
Example 6: Show that 0ʃ∞
e‒t sin t dt = 1/2
Solution:
We know that,
Example 7: Find L[t2 cos 2t]
Solution:
Example 8: Find L[te‒2t sin t]
Solution:
Example 9: Find the Laplace transform of [t
cos t sinh 2t]
Solution:
Example 10: Find (i) L [t2 cost]
(ii) L [t2 e‒1 cost]
Solution:
Example 11: Show that 0ʃ∞ e‒t
t cos t dt = 0
Solution:
Example 12: Find L [te‒t cosh t]
Solution:
Transforms and its Applications: UNIT 1: Laplace Transforms : Tag: Engineering mathematics, Maths : Laplace Transforms - Derivatives of Transform
Transforms and its Applications: UNIT 1: Laplace Transforms
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